Can I constrain an object size to Camera angle of view?
Actually, it can be done with both perspective and orthographic cameras. I would like to suggest the universal way, which needn't any reset on the current camera transformation:





For Perspective camera:


Enable Import Image as Planes and Simple Align in Addon list.
Import the target image as plane. When importing, Choose Dots/BU as Plane dimensions, and set the longest side of the image as Definition value. For example, if the image resolution is 1920 px x 1080 px, then you should type in 1920.
Set the render resolution the same as image, to set the correct perspective for camera.
With the plane selected, hold Shift to further select the camera, then find Simple Align tool in the Tool Shelf, click XYZ button to align both rotation and location of the plane to the camera. 
Select the plane only, then input like this: GZZ=-focal_length/sensor_size. For example, if the focal length of camera is 35, and sensor size is 32, then press GZZ=-35/32. Then you'll get what you want.




For Orthographic camera:


As above.
As above.
As above.
As above.
Select the plane only, then input like this:
GZZ=-ortho_scale/2. For example, if ortho_scale=8, you should input GZZ=-8/2. Then Tab into Edit Mode, S8.




P.S.: Doing math while transforming is only available in Blender 2.70+. However, you can do the math elsewhere out of Blender, and paste the result in.